Fluctuation Identities with Continuous Monitoring and Their Application to Price Barrier Options

30 Pages Posted: 5 Dec 2017

See all articles by Carolyn Phelan

Carolyn Phelan

University College London - Financial Computing and Analytics Group, Department of Computer Science

Daniele Marazzina

Polytechnic University of Milan - Department of Mathematics

Gianluca Fusai

Università del Piemonte Orientale Dipartimento di Studi per l'Economia e l'Impresa; Sir John Cass Business School - City, University of London

Guido Germano

University College London

Date Written: November 30, 2017

Abstract

We present a numerical scheme to calculate fluctuation identities for exponential L'evy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential L'evy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener-Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-$z$ domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

Keywords: Finance, Wiener-Hopf Factorisation, Hilbert Transform, Laplace Transform, Spectral Filter

JEL Classification: G12, C02

Suggested Citation

Phelan, Carolyn and Marazzina, Daniele and Fusai, Gianluca and Germano, Guido, Fluctuation Identities with Continuous Monitoring and Their Application to Price Barrier Options (November 30, 2017). Available at SSRN: https://ssrn.com/abstract=3080495 or http://dx.doi.org/10.2139/ssrn.3080495

Carolyn Phelan (Contact Author)

University College London - Financial Computing and Analytics Group, Department of Computer Science ( email )

Gower Street
London, WC1E 6BT
United Kingdom

Daniele Marazzina

Polytechnic University of Milan - Department of Mathematics ( email )

Via Bonardi, 9
Milano, MI 20133
Italy

Gianluca Fusai

Università del Piemonte Orientale Dipartimento di Studi per l'Economia e l'Impresa ( email )

Via Perrone, 18
Novara, 28100
Italy

HOME PAGE: http://https://upobook.uniupo.it/gianluca.fusai

Sir John Cass Business School - City, University of London ( email )

106 Bunhill Row
London, EC2Y 8HB
Great Britain

HOME PAGE: http:// www.cass.city.ac.uk/experts/G.Fusai

Guido Germano

University College London ( email )

Department of Computer Science
Gower Street
London, WC1E 6BT
United Kingdom
+44 20 3108 7105 (Phone)
+44 20 7387 1397 (Fax)

HOME PAGE: http://www.cs.ucl.ac.uk/people/G.Germano

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